Answered step by step
Verified Expert Solution
Question
1 Approved Answer
Consider the linear code $mathcal{Z}$ over $mathbb{F}_{2}=mathbb{Z}_{2}$ given by the generator matrix $$ G=left(begin{array}{cccccccc} 0 & 0 & 0 & 1 & 0 & 1
Consider the linear code $\mathcal{Z}$ over $\mathbb{F}_{2}=\mathbb{Z}_{2}$ given by the generator matrix $$ G=\left(\begin{array}{cccccccc} 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 0 & 0 & 1 & 0 & 0 & 0 & 0 & 11 1 & 0&1&1&1 & 0&1&1 W 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \end{array} ight) $$ 1. Determine the parameters $(n, M, d)$ and $[n, k, d]$ of the code. 2. Determine how many errors it can detect and how many errors it can correct. 3. Give examples when the code fails to correct and detect errors beyond its ability; for instance, if the code can correct at most a errors, show an example in which a 1 errors occur in which the received vector is decoded incorrectly. 4. Find a generator matrix for $\mathcal{Z}$ in systematic form. 5. Find a parity-check matrix for $\mathcal{Z}$. 6. Decode the following vectors using syndrome decoding: (a) $C01110111) ;$ (b) $(00111011) ;$ (c) $(01111001) $. CS.JG.011 Consider the linear code $\mathcal{Z}$ over $\mathbb{F}_{2}=\mathbb{Z}_{2}$ given by the generator matrix $$ G=\left(\begin{array}{cccccccc} 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 0 & 0 & 1 & 0 & 0 & 0 & 0 & 11 1 & 0&1&1&1 & 0&1&1 W 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \end{array} ight) $$ 1. Determine the parameters $(n, M, d)$ and $[n, k, d]$ of the code. 2. Determine how many errors it can detect and how many errors it can correct. 3. Give examples when the code fails to correct and detect errors beyond its ability; for instance, if the code can correct at most a errors, show an example in which a 1 errors occur in which the received vector is decoded incorrectly. 4. Find a generator matrix for $\mathcal{Z}$ in systematic form. 5. Find a parity-check matrix for $\mathcal{Z}$. 6. Decode the following vectors using syndrome decoding: (a) $C01110111) ;$ (b) $(00111011) ;$ (c) $(01111001) $. CS.JG.011
Step by Step Solution
There are 3 Steps involved in it
Step: 1
Get Instant Access to Expert-Tailored Solutions
See step-by-step solutions with expert insights and AI powered tools for academic success
Step: 2
Step: 3
Ace Your Homework with AI
Get the answers you need in no time with our AI-driven, step-by-step assistance
Get Started